Abstract

We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

1. Introduction

Differential equations of fractional order have been subjected to many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics, engineering, and so on. Recently, a large amount of literature was developed regarding the application of fractional differential equations in nonlinear dynamics (see, e.g., [1–11] and the references therein). Thus, a huge attention has been given to the solution of fractional ordinary differential equations, integral equations, and fractional partial differential equations of physical interest. As it is known, there exists no method that yields an exact solution for fractional differential equations. Various methods have been proposed in order to solve the fractional differential equations. These methods include the homotopy perturbation method [12–15], Adomian’s decomposition method [16–20], variation iteration method [12–14, 21–23], homotopy analysis method [24], differential transform method [25], operational matrices [26–28], and nonstandard finite difference scheme [29].

In this paper, we investigate the nonlinear system of fractional differential equations as
and the initial condition
where and . Also, are multivariable polynomial functions.

The structure of the paper is given later. In Section 2, we present some preliminaries and properties in fractional calculus and Bernstein polynomials. In Section 3, we make operational matrices for product, power, Caputo fractional derivative, and Riemann-Liouville fractional integral by BPs. In Section 4, we apply two methods for solving nonlinear system of fractional differential equations by BPs. In Section 5, numerical examples are simulated to demonstrate the high performance of the proposed method. Conclusions are presented in Section 6.

2. Basic Tools

In this section, we recall some basic definitions and properties of the fractional calculus and Bernstein polynomials.

Definition 1 (see [2, 7, 10]). The Riemann-Liouville fractional integral operator of order , of a function , is defined as
and for , , , , the fractional derivative of in the Caputo sense is defined as
where for we have
Also, if , and , then

Definition 2 (see [30]). The Bernstein polynomials (BPs) of th degree are defined on the interval as follows:

4. Solving System of Fractional Differential Equations

In this section, we use two methods for solving system of fractional differential equations. In the first method, we use the operational matrix for Caputo fractional derivative (OMCFD), and in the second method, we apply the operational matrix for Riemann-Liouville fractional integral (OMRLFI).

4.1. Solving the Problem by OMCFD

From (17) and (15) we can write
Therefore, problem (1) and (2) reduces to the following problem:
and the initial condition
Now, using Lemma 5 we can approximate all of the known functions in the system (19). Then, by using Lemma 7 and Corollaries 8 and 9, since functions are polynomial, we obtain the following approximations:
where .

Also, for each , by using tau method [33] we can generate algebraic equations from (19) and (21) as follows
and from (23) we set .

Finally, problem (1) and (2) has been reduced to the system of algebraic equations
The aforementioned system can be solved for by Newton’s iterative method. Then, we get the approximate value of the functions from (17).

4.2. Solving the Problem by OMRLFI

This method consists of two steps.

Step 1. Initial conditions are used to reduce a given initial-value problem to a problem with zero initial conditions. Therefore we have a modified system, incorporating the initial values.

Step 2. The BPs operational matrix of Riemann-Liouville fractional integral is used to transform the problem into a system of algebraic equations.

Substituting (24) in (1) and (2), we have the following system:
and the initial condition
where and are multivariable polynomial functions. We use the following approximation:
where are unknown vectors. From (7), (27), and Theorem 11, we can write
So, by (27) and (28), problem (25) and (26) reduces to the following problem:
As we saw in the previous section, we can obtain the following approximations:
where . So, from (29) and (30) we have
Therefore, we have reduced problem (1) and (2) to the system of algebraic equations as follows:
where this system can be solved for by Newton’s iterative method. Finally we obtain the approximate of the functions by

5. Examples

To demonstrate the applicability and to validate the numerical scheme, we apply the present method for the following examples.

Example 12. Consider the following linear system of fractional differential equations [24, 25]:
with initial condition
For this problem we have the exact solution in the case of as
We solved this problem by OMCFD and OMRLFI. Figures 1 and 2 show the approximate solutions of and , respectively, as a function of time for , for different values of , . The results show that numerical solutions are in good agreement with each other, in both methods. Also, these figures show that as , approach close to 1, the numerical solutions approach to the solutions for as expected. In Figures 3 and 4, we see the absolute error of both methods, for , . In these figures, we can see that obtained results using the presented methods agree well with the analytical solutions for .

Figure 1: Approximate solutions of for and different values of , by OMCFD and OMRLFI in Example 12.

Figure 2: Approximate solutions of for and different values of , by OMCFD and OMRLFI in Example 12.

Figure 3: Plot of absolute error function for and by OMCFD and OMRLFI in Example 12.

Figure 4: Plot of absolute error function for and by OMCFD and OMRLFI in Example 12.

Example 13. Let us consider the following nonlinear fractional system [24] as follows:
such that

The exact solution of this system, when , is
Figures 5 and 6 show the approximate solutions of and , respectively, for different values of , by OMCFD and OMRLFI. We conclude that as , approach close to 1, the numerical solutions approach solutions for as expected. Furthermore, in both methods, the results agree well with each other. Figures 7 and 8 show that, the absolute error of obtained results for and using OMCFD and OMRLFI is in good agreement with the exact solution.

Figure 5: Approximate solutions of for and different values of , by OMCFD and OMRLFI in Example 13.

Figure 6: Approximate solutions of for and different values of , by OMCFD and OMRLFI in Example 13.

Figure 7: Plot of absolute error function for and by OMCFD and OMRLFI in Example 13.

Figure 8: Plot of absolute error function for and by OMCFD and OMRLFI in Example 13.

Example 14. Consider the nonlinear system of fractional differential equations [24]:
with the initial conditions given by
The exact solution of this system, when , becomes
We can see the approximate solutions of and , by OMCFD and OMRLFI for and different values of , and , in Figures 9, 10, and 11. These figures show that, when , , and approach close to 1, the numerical solutions approach the solutions for as expected. In Figures 9–11, we observe that results of OMCFD and OMRLFI overlap. In Figures 12, 13, and 14, we see the absolute error of the obtained results for and in both methods.

Figure 9: Approximate solutions of for and different values of , , by OMCFD and OMRLFI in Example 14.

Figure 10: Approximate solutions of for and different values of , , by OMCFD and OMRLFI in Example 14.

Figure 11: Approximate solutions of for and different values of , , by OMCFD and OMRLFI in Example 14.

Figure 12: Plot of absolute error function for and by OMCFD and OMRLFI in Example 14.

Figure 13: Plot of absolute error function for and by OMCFD and OMRLFI in Example 14.

Figure 14: Plot of absolute error function for and by OMCFD and OMRLFI in Example 14.

6. Conclusion

In this paper, we get operational matrices of the product, Caputo fractional derivative, and Riemann-Liouville fractional integral by Bernstein polynomials. Then by using these matrices, we proposed two methods that reduced the nonlinear systems of fractional differential equations to the two system of algebraic equations that can be solved easily. Finally, numerical examples are simulated to demonstrate the high performance of the proposed method. We saw that the results of both methods were in good agreement with each other, and the classical solutions were recovered when the order of the fractional derivative goes to 1.